IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v7y2019i11p1100-d286750.html
   My bibliography  Save this article

A Fractional-Order Predator–Prey Model with Ratio-Dependent Functional Response and Linear Harvesting

Author

Listed:
  • Agus Suryanto

    (Department of Mathematics, Faculty of Mathematics and Natural Sciences, University of Brawijaya, Malang 65145, Indonesia)

  • Isnani Darti

    (Department of Mathematics, Faculty of Mathematics and Natural Sciences, University of Brawijaya, Malang 65145, Indonesia)

  • Hasan S. Panigoro

    (Department of Mathematics, Faculty of Mathematics and Natural Sciences, University of Brawijaya, Malang 65145, Indonesia
    Department of Mathematics, Faculty of Mathematics and Natural Sciences, State University of Gorontalo, Gorontalo 96128, Indonesia)

  • Adem Kilicman

    (Department of Mathematics, Faculty of Science, Universiti Putra Malaysia, 43400 UPM Serdang, Selangor, Malaysia)

Abstract

We consider a model of predator–prey interaction at fractional-order where the predation obeys the ratio-dependent functional response and the prey is linearly harvested. For the proposed model, we show the existence, uniqueness, non-negativity and boundedness of the solutions. Conditions for the existence of all possible equilibrium points and their stability criteria, both locally and globally, are also investigated. The local stability conditions are derived using the Magtinon’s theorem, while the global stability is proven by formulating an appropriate Lyapunov function. The occurrence of Hopf bifurcation around the interior point is also shown analytically. At the end, we implemented the Predictor–Corrector scheme to perform some numerical simulations.

Suggested Citation

  • Agus Suryanto & Isnani Darti & Hasan S. Panigoro & Adem Kilicman, 2019. "A Fractional-Order Predator–Prey Model with Ratio-Dependent Functional Response and Linear Harvesting," Mathematics, MDPI, vol. 7(11), pages 1-13, November.
    • Handle: RePEc:gam:jmathe:v:7:y:2019:i:11:p:1100-:d:286750
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/7/11/1100/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/7/11/1100/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Hamdan, Nur ’Izzati & Kilicman, Adem, 2018. "A fractional order SIR epidemic model for dengue transmission," Chaos, Solitons & Fractals, Elsevier, vol. 114(C), pages 55-62.
    2. Sung Kyu Choi & Bowon Kang & Namjip Koo, 2014. "Stability for Caputo Fractional Differential Systems," Abstract and Applied Analysis, Hindawi, vol. 2014, pages 1-6, January.
    3. Moustafa, Mahmoud & Mohd, Mohd Hafiz & Ismail, Ahmad Izani & Abdullah, Farah Aini, 2018. "Dynamical analysis of a fractional-order Rosenzweig–MacArthur model incorporating a prey refuge," Chaos, Solitons & Fractals, Elsevier, vol. 109(C), pages 1-13.
    4. Agus Suryanto & Isnani Darti & Syaiful Anam, 2017. "Stability Analysis of a Fractional Order Modified Leslie-Gower Model with Additive Allee Effect," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 2017, pages 1-9, May.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Uddin, Md. Jasim & Rana, Sarker Md. Sohel & Işık, Seval & Kangalgil, Figen, 2023. "On the qualitative study of a discrete fractional order prey–predator model with the effects of harvesting on predator population," Chaos, Solitons & Fractals, Elsevier, vol. 175(P1).

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Ghosh, Uttam & Pal, Swadesh & Banerjee, Malay, 2021. "Memory effect on Bazykin’s prey-predator model: Stability and bifurcation analysis," Chaos, Solitons & Fractals, Elsevier, vol. 143(C).
    2. Aliyu, Murtala Bello & Mohd, Mohd Hafiz, 2021. "The interplay between mutualism, competition and dispersal promotes species coexistence in a multiple interactions type system," Ecological Modelling, Elsevier, vol. 452(C).
    3. Huang, Chengdai & Liu, Heng & Chen, Xiaoping & Zhang, Minsong & Ding, Ling & Cao, Jinde & Alsaedi, Ahmed, 2020. "Dynamic optimal control of enhancing feedback treatment for a delayed fractional order predator–prey model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 554(C).
    4. Zhou, Jiaying & Ye, Yong & Arenas, Alex & Gómez, Sergio & Zhao, Yi, 2023. "Pattern formation and bifurcation analysis of delay induced fractional-order epidemic spreading on networks," Chaos, Solitons & Fractals, Elsevier, vol. 174(C).
    5. Majumdar, Prahlad & Mondal, Bapin & Debnath, Surajit & Ghosh, Uttam, 2022. "Controlling of periodicity and chaos in a three dimensional prey predator model introducing the memory effect," Chaos, Solitons & Fractals, Elsevier, vol. 164(C).
    6. Chunning Song & Yu Zhang & Qijin Ling & Shaogeng Zheng, 2022. "Joint Estimation of SOC and SOH for Single-Flow Zinc–Nickel Batteries," Energies, MDPI, vol. 15(13), pages 1-16, June.
    7. Idris Ahmed & Chanakarn Kiataramkul & Mubarak Muhammad & Jessada Tariboon, 2024. "Existence and Sensitivity Analysis of a Caputo Fractional-Order Diphtheria Epidemic Model," Mathematics, MDPI, vol. 12(13), pages 1-18, June.
    8. Abdullahi, Auwal, 2021. "Modelling of transmission and control of Lassa fever via Caputo fractional-order derivative," Chaos, Solitons & Fractals, Elsevier, vol. 151(C).
    9. Malik, Hafiz Abid Mahmood & Abid, Faiza & Wahiddin, Mohamed Ridza & Waqas, Ahmad, 2021. "Modeling of internal and external factors affecting a complex dengue network," Chaos, Solitons & Fractals, Elsevier, vol. 144(C).
    10. Mohd, Mohd Hafiz & Md. Noorani, Mohd Salmi, 2021. "Local dispersal, trophic interactions and handling times mediate contrasting effects in prey-predator dynamics," Chaos, Solitons & Fractals, Elsevier, vol. 142(C).
    11. Yadav, Ram Prasad & Renu Verma,, 2020. "A numerical simulation of fractional order mathematical modeling of COVID-19 disease in case of Wuhan China," Chaos, Solitons & Fractals, Elsevier, vol. 140(C).
    12. Das, Meghadri & Samanta, G.P., 2020. "A delayed fractional order food chain model with fear effect and prey refuge," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 178(C), pages 218-245.
    13. Jajarmi, Amin & Baleanu, Dumitru, 2018. "A new fractional analysis on the interaction of HIV with CD4+ T-cells," Chaos, Solitons & Fractals, Elsevier, vol. 113(C), pages 221-229.
    14. Tuan Hoang, Manh & Nagy, A.M., 2019. "Uniform asymptotic stability of a Logistic model with feedback control of fractional order and nonstandard finite difference schemes," Chaos, Solitons & Fractals, Elsevier, vol. 123(C), pages 24-34.
    15. Sekerci, Yadigar, 2020. "Climate change effects on fractional order prey-predator model," Chaos, Solitons & Fractals, Elsevier, vol. 134(C).
    16. Mangal, Shiv & Misra, O.P. & Dhar, Joydip, 2023. "Fractional-order deterministic epidemic model for the spread and control of HIV/AIDS with special reference to Mexico and India," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 210(C), pages 82-102.
    17. Zhang, Xuncai & Wang, Shida & Zhao, Kai & Wang, Yanfeng, 2023. "A salp swarm algorithm based on Harris Eagle foraging strategy," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 203(C), pages 858-877.
    18. Yousef, Fatma Bozkurt & Yousef, Ali & Maji, Chandan, 2021. "Effects of fear in a fractional-order predator-prey system with predator density-dependent prey mortality," Chaos, Solitons & Fractals, Elsevier, vol. 145(C).
    19. Li, Hong-Li & Kao, Yonggui & Hu, Cheng & Jiang, Haijun & Jiang, Yao-Lin, 2021. "Robust exponential stability of fractional-order coupled quaternion-valued neural networks with parametric uncertainties and impulsive effects," Chaos, Solitons & Fractals, Elsevier, vol. 143(C).
    20. Moustafa, Mahmoud & Mohd, Mohd Hafiz & Ismail, Ahmad Izani & Abdullah, Farah Aini, 2018. "Dynamical analysis of a fractional-order Rosenzweig–MacArthur model incorporating a prey refuge," Chaos, Solitons & Fractals, Elsevier, vol. 109(C), pages 1-13.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:7:y:2019:i:11:p:1100-:d:286750. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.